Introduction to Matrices for Engineers


 Diana Boyd
 5 years ago
 Views:
Transcription
1 Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g The above matrix is a 4 matrix, i.e. it has three columns and four rows. 1.1 Wh use Matrices? We use matrices in mathematics and engineering because often we need to deal with several variables at once eg the coordinates of a point in the plane are written x, or in space as x,, z and these are often written as column matrices in the form: x and x z It turns out that man operations that are needed to be performed on coordinates of points are linear operations and so can be organized in terms of rectangular arras of numbers, matrices. Then we find that matrices themselves can under certain conditions be added, subtracted and multiplied so that there arises a whole new set of algebraic rules for their manipulation. In general, an n m matrix A looks like: A a 11 a 1 a 1... a 1,m 1 a 1,m a 1 a a... a,m 1 a,m a 1 a a... a,m 1 a,m a n 1,1 a n 1, a n 1,... a n 1,m 1 a n 1,m a n,1 a n, a n,... a n,m 1 a n,m Here, the entries are denoted a ij ; e.g. in Example we have a 11 1, a, etc. Capital letters are usuall used for the matrix itself. 1. Dimension In the above matrix A, the numbers n and m are called the dimensions of A. Based on original lecture notes on matrices b Lewis Pirnie 1
2 Introduction to Matrices 1. Addition It is possible to add two matrices together, but onl if the have the same dimensions. In which case we simpl add the corresponding entries: If two matrices don t have the same size dimensions then the can t be added, or we sa the sum is not defined. 1.4 Example is undefined Multipling Matrices.1 Wh should we want to? We can motivate the process b looking at rotations of points in the plane. Consider the point P on the xaxis at x x, 0, so it has coordinates x 0 Now, let P be the point obtained b rotating P round the origin in an anticlockwise direction through angle θ, keeping the distance from the origin which is actuall the square root of the sum of squares of the coordinates constant. What are the coordinates of the point P? It is not difficult trigonometr to work out that these are: x cos θ x sin θ Next consider the point Q on the axis with coordinates 0 and rotate this point round the origin in an anticlockwise direction through angle θ, keeping the distance from the origin constant. We find that the new coordinates are: sin θ cos θ Now the paoff, it is actuall a matrix that does the rotating here because the operation is linear on the coordinate components. We can express the rotation of an point with coordinates x, as the following matrix equation: cos θ sin θ sin θ cos θ x x cos θ sin θ x sin θ + cos θ Can ou guess what will be the matrix for rotation in a clockwise direction?
3 C.T.J. Dodson. Exercise on Trigonometr 1. Three common right angled triangles can be used to obtain the following values for the trigonometric functions: 0 π 6 45 π 4 60 π sin π 6 1 sin π 4 1 sin π cos π 6 cos π 4 1 cos π 1 tan π 6 1 tan π 4 1 tan π. Obtain the rotation matrices explicitl for rotations of θ ±0, ±45, ±60, ±90, ±180.. Rules When multipling matrices, keep the following in mind: la the first row of the first matrix on top of the first column of the second matrix; onl if the are both of the same size can ou proceed. The rule for multipling is: go across the first matrix, and down the second matrix, multipling the corresponding entries, and adding the products. This new number goes in the new matrix in position of the row of the first matrix, and the column of the second matrix. For example: Smbolicall, if we have the matrices A and B as follows: A a 11 a 1 a 1... a 1,m a 1 a a... a,m a 1 a a... a,m a n 1,1 a n 1, a n 1,... a n 1,m a n,1 a n, a n,... a n,m, B b 11 b 1 b 1... b 1,q b 1 b b... b,q b 1 b b... b,q b p 1,1 b p 1, b p 1, b p 1,q 1 b p 1,q b p,1 b p, b p,... b p,q then the product AB is given b: m i1 a m 1ib i1 i1 a m 1ib i... i1 a m 1ib iq i1 a m ib i1 i1 a m ib i... i1 a ib iq m i1 a m nib i i1 a m nib i... i1 a nib iq where m i1 a 1ib i1 stands for a 11 b 11 + a 1 b 1 + a 1 b a 1n b n1, etc. Note that we must have m p, i.e. the number os columns in the first matrix must equal the number of rows of the second; otherwise, we sa the product is undefined.
4 4 Introduction to Matrices.4 Example We multipl the following matrices: 1 0 i ii is undefined iii iv v vi We see from the examples: i Product of matrix with a matrix is a matrix. iii Product of matrix with a matrix is a matrix. iv Product of matrix with a 4 matrix is a 4 matrix. v Product of matrix with a 1 matrix is a 1 matrix. vi Product of matrix with a 4 matrix is a 4 matrix. Note that the two middle numbers must be the same if the product is defined; and then the dimensions of the answer is just the two outer numbers. Thus, the product of an n m matrix with a m q matrix is an n q matrix. Scalar Multiplication There is another tpe of multiplication involving matrices called scalar multiplication. This means just multipling each entr of the matrix b a number. For example: Rules There are some rules which matrix addition and multiplication obe: Associative A + B + C A + B + C Commutative A + B B + A
5 C.T.J. Dodson 5 Distributive Distributive Associative Non Commutative Moving Constants AB + C AB + AC A + BC AC + BC ABC ABC AB BA usuall AλB λab Assuming that the sums and products are defined in all cases.. Example Let A , B 1 , C Consider the following: AB BA Now, AB BA, and we see that two matrices are not the same if the are multiplied the other wa around. Also consider: AC BC AB + BC A + B A + BC and we notice that A + BC AC + BC as required Transpose Another operation on matrices is the transpose. This just reverses the rows and columns, or equivalentl, reflects the matrix along the leading diagonal. The transpose of A is normall written A t thus a 11 a 1 a 1... a 1,m 1 a 1,m a 1 a a... a,m 1 a,m A a 1 a a... a,m 1 a,m , a n 1,1 a n 1, a n 1,... a n 1,m 1 a n 1,m a n,1 a n, a n,... a n,m 1 a n,m
6 6 Introduction to Matrices A t a 11 a 1 a 1... a m 1,1 a m,1 a 1 a a... a m 1, a m, a 1 a a... a m 1, a m, a 1,n 1 a,n 1 a,n 1... a m 1,n 1 a m,n 1 a 1,n a,n a,n... a m 1,n a m,n Note that the transpose of a n m matrix is a m n matrix. 4.1 Example As an example of the transpose: A , A t Square Matrices Given a number n, n n matrices have ver special properties. Note that if we have two n n matrices, the product is defined and will also be an n n matrix. 5.1 The Identit Matrix There also exists a special matrix, known as the identit, I n n : I , I , I , etc. which has 1 s on the main diagonal, and 0 s everwhere else. This matrix has the propert that, given an n n matrix A: AI n n A and I n n A A i.e. multipling b the identit on either side doesn t change the matrix. This is similar to the propert of 1 when multipling numbers. We usuall abbreviate I n n to just I when it s obvious what n is. 5. Example Let A 1 1 0, I I 0 1 Now, we check the properties of the identit: as required. AI IA
7 C.T.J. Dodson 6 Determinants One of the most important properties of square matrices is the determinant. This is a number obtained from the entries. 6.1 Determinant of a Matrix a b Let A. Then, the determinant of A, denoted det A or A is given b ad bc. c d 6. Example det det Before we go on to larger matrices, we need to define minors. 6. Minors Let A be the n n matrix A a 11 a 1 a 1... a 1,m 1 a 1,m a 1 a a... a,m 1 a,m a 1 a a... a,m 1 a,m a n 1,1 a n 1, a n 1,... a n 1,m 1 a n 1,m a n,1 a n, a n,... a n,m 1 a n,m Then, the minor m ij, for each i, j, is the determinant of the n 1 n 1 matrix obtained b deleting the i th row and the j th column. For example, in the above notation: 6.4 Example m 11 det m 1 det We compute all the minors of A a a... a,m 1 a,m a a... a,m 1 a,m a n 1, a n 1,... a n 1,m 1 a n 1,m a n, a n,... a n,m 1 a n,m a 1 a 1... a 1,m 1 a 1,m a a... a,m 1 a,m a n 1, a n 1,... a n 1,m 1 a n 1,m a n, a n,... a n,m 1 a n,m m m 1 m m m m
8 8 Introduction to Matrices m m m Minors and Cofactors The numbers called cofactors are almost the same as minors, except some have a minus sign in accordance with the following pattern: The best wa to remember this is as an alternating or chessboard pattern. The cofactors from the previous example are: c 11 m 11 8 c 1 m 1 15 c 1 m 1 0 c 1 m 1 c m 9 c m 5 c 1 m 1 c m 6 c m 8 Determinant of a Matrix In order to calculate the determinant of a matrix, choose an row or column. Then, multipl each entr b its corresponding cofactor, and add the three products. This gives the determinant..1 Example Letting A 0 4 as before, we compute the determinant using the top row: det A a 11 c 11 + a 1 c 1 + a 1 c Suppose, we use the second column instead: det A a 1 c 1 + a c + a c It doesn t matter which row or column is used, but the top row is normal. Note that it is not necessar to work out all the minors or cofactors, just three.. Example Let B We compute the determinant of B: det
9 C.T.J. Dodson 9 8 Determinant of an n n Matrix The procedure for larger matrices is exactl the same as for a matrix: choose a row or column, multipl the entr b the corresponding cofactor, and add them up. But of course each minor is itself the determinant of an n 1 n 1 matrix, so for example, in a 4 4 determinant, it is first necessar to do four determinants quite a lot of work! 9 Inverses Let A be an n n matrix, and let I be the n n identit matrix. Sometimes, there exists a matrix A 1 called the inverse of A with the propert: A A 1 I A 1 A In this section, we demonstrate a method for finding inverses. 9.1 Inverse of a Matrix a b Let A. Then, the inverse of A, A c d 1 is given b: A 1 1 d b ad bc c a To check, we multipl: A 1 1 d b a b 1 da bc db bd A ad bc c a c d ad bc ca + ac cb + ad 1 ad bc ad bc 0 ad bc 0 1 In a similar fashion we could show that A A 1 I. Of course, the inverse could also be written A 1 1 d b det A c a Note, that if det A 0, then we have a division b zero, which we can t do. In this situation there is no inverse of A. 9. Inverse of and higher Matrices Recall the definition of a minor from Section 4..: given an n n matrix A, the minor m ij is the determinant of the n 1 n 1 matrix obtained b omitting the i th row and the j th column. 9. Example Let A We calculate the minors: m m 1 m m m m m
10 10 Introduction to Matrices Recall also the pattern of + and signs from which we obtain the cofactors: Now, we put the minors into a matrix and change their signs according to the pattern to get the matrix of cofactors: The next stage is take the transpose: and finall we must divide b the determinant, which is, from Example 4..8: A This shows how to calculate the inverse of a matrix. We check the result: A 1 A as required The same procedure works for n n matrices I II III IV Work out the minors. Put in the signs to form the cofactors. Take the transpose. Divide b the determinant. Furthermore, an n n matrix has an inverse if and onl if the determinant is not zero. So, it s a good idea to calculate the determinant first, just to check whether the rest of the procedure is necessar. 10 Linear Sstems We discuss one ver important application of finding inverses of matrices Simultaneous Equations Often, when solving problems in mathematics, we need to solve simultaneous equations, e.g. something like: x + 5x +
11 C.T.J. Dodson 11 from which we would obtain x and 1. The process we have used up until now is a little mess: we combine the equations to tr and eliminate one of the unknown variables. There is a more sstematic wa using matrices. We can write the equations in a slightl different wa: x + 5x + Now we can check that the first matrix is equal to the product: x + 1 x 5x + 5 and so altogether we have a matrix equation: 1 x 5 The next stage is to use the inverse of the matrix, so let s calculate that now. Let A 1 5, then A Now, we take the matrix equation above, and multipl b A x Then, doing the multiplication: x x x 1 and so x and 1, as required. So, provided we can work out the inverse of the matrix of coefficients, we can solve simultaneous equations. 11 Larger Sstems The same thing works with equations in x, and z. Suppose we have x + + z 1 z x + 8z Then, the matrix form is x z 1 Now, we denote the matrix b A, and calculate the inverse of A. The minors are as follows:
12 1 Introduction to Matrices m m m m m m m 110 m m 1 0 Recall the chessboard pattern: So we have the following matrix of cofactors: We can calculate the determinant b taking an row or column, and multipling the original matrix entr b its corresponding cofactor, and then adding let s choose the top row: det A and so the determinant is, and so we will be able to find the inverse. From the matrix of cofactors, we take the transpose, and then divide b the determinant to get A 1 : A Now, we return to solving the simultaneous equations, where we had: 1 x z Multipling both sides on the left b A 1, we have: x z and we know that A 1 A I, so x z x z and so we get x 64, 1 and z. It s a ver good idea to check calculations like this! x + + z z 1 9 x + 8z as required.
13 C.T.J. Dodson 1 1 Appendix 1.1 Scientific Wordprocessing with L A TEX This pdf document with its hperlinks was created using L A TEX which is the standard free mathematical wordprocessing package; more information can be found via the webpage: 1. Computer Algebra Methods The computer algebra package Mathematica can be used to manipulate and invert matrices. See: for beginning Mathematica. Similarl, Maple and Matlab also can be used for working with matrices.
MAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is nonnegative
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationGCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook
Version 36 klm GCE Mathematics (636) Further Pure unit 4 (MFP4) Textbook The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 364473 and a
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More information7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a 0 has a reciprocal b written as a or such that a ba = ab =. Similarly a square matrix A may have an inverse B = A where AB =
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationSECTION 74 Algebraic Vectors
74 lgebraic Vectors 531 SECTIN 74 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors
More information1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationComplex Numbers. w = f(z) z. Examples
omple Numbers Geometrical Transformations in the omple Plane For functions of a real variable such as f( sin, g( 2 +2 etc ou are used to illustrating these geometricall, usuall on a cartesian graph. If
More informationChapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6
Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a
More informationSection 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationCOMPLEX STRESS TUTORIAL 3 COMPLEX STRESS AND STRAIN
COMPLX STRSS TUTORIAL COMPLX STRSS AND STRAIN This tutorial is not part of the decel unit mechanical Principles but covers elements of the following sllabi. o Parts of the ngineering Council eam subject
More informationAddition and Subtraction of Vectors
ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationMatrices 2. Solving Square Systems of Linear Equations; Inverse Matrices
Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationTypical Linear Equation Set and Corresponding Matrices
EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
More informationMATHEMATICS FOR ENGINEERS BASIC MATRIX THEORY TUTORIAL 2
MATHEMATICS FO ENGINEES BASIC MATIX THEOY TUTOIAL This is the second of two tutorials on matrix theory. On completion you should be able to do the following. Explain the general method for solving simultaneous
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationMath, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.
Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical
More informationSolving simultaneous equations using the inverse matrix
Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix
More information28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z
28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition
More informationName: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
More informationsin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj
Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More information8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes
Solution by Inverse Matrix Method 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix algebra allows us
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationMA106 Linear Algebra lecture notes
MA106 Linear Algebra lecture notes Lecturers: Martin Bright and Daan Krammer Warwick, January 2011 Contents 1 Number systems and fields 3 1.1 Axioms for number systems......................... 3 2 Vector
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationVectors Math 122 Calculus III D Joyce, Fall 2012
Vectors Math 122 Calculus III D Joyce, Fall 2012 Vectors in the plane R 2. A vector v can be interpreted as an arro in the plane R 2 ith a certain length and a certain direction. The same vector can be
More informationQuestion 2: How do you solve a matrix equation using the matrix inverse?
Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients
More information6. Vectors. 1 20092016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is socalled because when the scalar product of
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationLecture notes on linear algebra
Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More informationLinear Algebraic Equations, SVD, and the PseudoInverse
Linear Algebraic Equations, SVD, and the PseudoInverse Philip N. Sabes October, 21 1 A Little Background 1.1 Singular values and matrix inversion For nonsmmetric matrices, the eigenvalues and singular
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationF Matrix Calculus F 1
F Matrix Calculus F 1 Appendix F: MATRIX CALCULUS TABLE OF CONTENTS Page F1 Introduction F 3 F2 The Derivatives of Vector Functions F 3 F21 Derivative of Vector with Respect to Vector F 3 F22 Derivative
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationIntroduction to Matrices
Introduction to Matrices Tom Davis tomrdavis@earthlinknet 1 Definitions A matrix (plural: matrices) is simply a rectangular array of things For now, we ll assume the things are numbers, but as you go on
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 51 Orthonormal
More informationConnecting Transformational Geometry and Transformations of Functions
Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationUsing row reduction to calculate the inverse and the determinant of a square matrix
Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible
More informationa.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a
Bellwork a.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationSome Lecture Notes and InClass Examples for PreCalculus:
Some Lecture Notes and InClass Examples for PreCalculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax
More informationUnit 18 Determinants
Unit 18 Determinants Every square matrix has a number associated with it, called its determinant. In this section, we determine how to calculate this number, and also look at some of the properties of
More informationMATH 551  APPLIED MATRIX THEORY
MATH 55  APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More informationTo Evaluate an Algebraic Expression
1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle  in particular the basic equation representing
More informationTHE COMPLEX EXPONENTIAL FUNCTION
Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula
More informationChapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors
Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationLinear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development
MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationAP Physics  Vector Algrebra Tutorial
AP Physics  Vector Algrebra Tutorial Thomas Jefferson High School for Science and Technology AP Physics Team Summer 2013 1 CONTENTS CONTENTS Contents 1 Scalars and Vectors 3 2 Rectangular and Polar Form
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More information